Generalized Schrodinger Equation Research Articles

From Generalized Hamilton Principle to Generalized Schrodinger Equation

The Hamilton principle is a variation principle describing the isolated and conservative systems, its Lagrange function is the difference between kinetic energy and potential energy. By Feynman path integration, we can obtain the standard Schrodinger equation. In this paper, we have given the generalized Hamilton principle, which can describe the heat exchange system, and the nonconservative force system. On this basis, we have further given their generalized Lagrange functions and Hamilton functions. With the Feynman path integration, we have given the generalized Schrodinger equation of nonconservative force system and the heat exchange system.

Quantum theory-based physical model of the human body in TCM

In the study, a quantum resonant cavity model based on wave-particle duality was proposed for the explanation of the dynamic processes of essence, vigor, and spirit in the human body in traditional Chinese medicine (TCM). It is assumed that there is a macro human order parameter (wave function), and its dynamics are governed by a macro potential field reflecting influences from heaven, earth, and society, and satisfy the generalized Schrodinger equation. This proposed model was applied in the study to interpret basic concepts of human body in TCM, with an aim to unfold the TCM development in the future.

Novel Optical Soliton Waves in Metamaterials with Parabolic Law of Nonlinearity via the IEFM and ISEM

Here, the miscellaneous soliton solutions of the generalized nonlinear Schrödinger equation are considered that describe the model of few-cycle pulse propagation in metamaterials with parabolic law of nonlinearity. The novel analytical wave solutions to the mentioned nonlinear equation in the sense of the nonlinear ordinary differential transform equation are obtained. The techniques are the improved exp − Γ ϖ function method and the improved simple equation method. The nonlinear ordinary transform to concern the generalized Schrodinger equation to convert it for a solvable integer-order differential equation is used. After the successful implementation of the presented methods, the exact solitary wave solutions in the form of trigonometric, rational, and hyperbolic functions are obtained. Hence, the presented methods are relatable and efficient to solve nonlinear problems in mathematical physics.

Dynamics of position-dependent mass particle in crystal lattices microstructures

Abstract In this study, we have combined the new concept of generalized momentum operator in quantum mechanics with the framework of position-dependent mass which plays a crucial role in nanomaterials sciences and technologies. We have derived the generalized Schrodinger equation and we have discussed several of its consequences. The generalized Schrodinger equation may describe a large number of quantum mechanical problems related to semiconductors and materials sciences. We have analyzed several problems depending on the forms of the position-dependent mass and the generalized momentum operator. For particular cases, it was observed that free particles behave as oscillators with energy levels less than the standard result which corresponds to the harmonic oscillator. The generalized Schrodinger equation is suitable also to describe quantum damping oscillators. We have also analyzed the dynamics of enormously slowly growing exponentially position-dependent electrons mass moving in a weak periodic potential and we have derived the consequent energy levels in particular for large lattice's length. The corresponding energy levels are enhanced and separated into allowed bands divided by band gaps. This result is in agreement with recent observations which are related to the presence of weak disorder and nonlinearities. Further details are analyzed accordingly.

Derivation of a generalized Schrödinger equation for dark matter halos from the theory of scale relativity

Abstract Using Nottale’s theory of scale relativity, we derive a generalized Schrodinger equation applying to dark matter halos. This equation involves a logarithmic nonlinearity associated with an effective temperature and a source of dissipation. Fundamentally, this wave equation arises from the nondifferentiability of the trajectories of the dark matter particles whose origin may be due to ordinary quantum mechanics, classical ergodic (or almost ergodic) chaos, or to the fractal nature of spacetime at the cosmic scale. The generalized Schrodinger equation involves a coefficient D , possibly different from ħ ∕ 2 m (where ħ is the Planck constant and m the mass of the particles), whose value for dark matter halos is D = 1 . 02 × 1 0 23 m2/s. This model is similar to the Bose–Einstein condensate dark matter model except that it does not require the dark matter particle to be ultralight. It can accommodate any type of particles provided that they have nondifferentiable trajectories. We suggest that the cold dark matter crisis may be solved by the fractal (nondifferentiable) structure of spacetime at the cosmic scale, or by the chaotic motion of the particles on a very long timescale, instead of ordinary quantum mechanics. The equilibrium states of the generalized Schrodinger equation correspond to configurations with a core-halo structure. The quantumlike potential generates a solitonic core that solves the cusp problem of the classical cold dark matter model. The logarithmic nonlinearity accounts for the presence of an isothermal halo that leads to flat rotation curves (it also accounts for the isothermal core of large dark matter halos). The damping term ensures that the system relaxes towards an equilibrium state. This property is guaranteed by an H -theorem satisfied by a Boltzmann-like free energy functional. In our approach, the temperature and the friction arise from a single formalism. They correspond to the real and imaginary parts of the complex friction coefficient present in the scale covariant equation of dynamics that is at the basis of Nottale’s theory of scale relativity. They may be the manifestation of a cosmic aether or the consequence of a process of violent relaxation and gravitational cooling on a coarse-grained scale.

Infinitely many solutions for Kirchhoff equations with sign-changing potential and Hartree nonlinearity

This paper is concerned with the following Kirchhoff-type equations: $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\big (a+b\int _{\mathbb {R}^{3}}|\nabla u|^{2}\mathrm {d}x\big )\Delta u+ V(x)u+\mu \phi |u|^{p-2}u=f(x, u)+g(x,u), &{} \text{ in } \mathbb {R}^{3},\\ (-\Delta )^{\frac{\alpha }{2}} \phi = \mu |u|^{p}, &{} \text{ in } \mathbb {R}^{3},\\ \end{array} \right. \end{aligned}$$ where $$a>0,~b,~\mu \ge 0$$ are constants, $$\alpha \in (0,3)$$ , $$p\in [2,3+2\alpha )$$ , the potential V(x) may be unbounded from below and $$\phi |u|^{p-2}u$$ is a Hartree-type nonlinearity. Under some mild conditions on V(x), f(x, u) and g(x, u), we prove that the above system has infinitely many nontrivial solutions. Specially, our results cover the general Schrodinger equations, the Kirchhoff equations and the Schrodinger–Poisson system.

Derivation of a generalized Schrödinger equation from the theory of scale relativity

Using Nottale’s theory of scale relativity relying on a fractal space-time, we derive a generalized Schrodinger equation taking into account the interaction of the system with the external environment. This equation describes the irreversible evolution of the system towards a static quantum state. We first interpret the scale-covariant equation of dynamics stemming from Nottale’s theory as a hydrodynamic viscous Burgers equation for a potential flow involving a complex velocity field and an imaginary viscosity. We show that the Schrodinger equation can be directly obtained from this equation by performing a Cole-Hopf transformation equivalent to the WKB transformation. We then introduce a friction force proportional and opposite to the complex velocity in the scale-covariant equation of dynamics in a way that preserves the local conservation of the normalization condition. We find that the resulting generalized Schrodinger equation, or the corresponding fluid equations obtained from the Madelung transformation, involve not only a damping term but also an effective thermal term. The friction coefficient and the temperature are related to the real and imaginary parts of the complex friction coefficient in the scale-covariant equation of dynamics. This may be viewed as a form of fluctuation-dissipation theorem. We show that our generalized Schrodinger equation satisfies an H-theorem for the quantum Boltzmann free energy. As a result, the probability distribution relaxes towards an equilibrium state which can be viewed as a Boltzmann distribution including a quantum potential. We propose to apply this generalized Schrodinger equation to dark matter halos in the Universe, possibly made of self-gravitating Bose-Einstein condensates.

Theory of Stochastic Schrödinger Equation in Complex Vector Space

A generalized Schrodinger equation containing correction terms to classical kinetic energy, has been derived in the complex vector space by considering an extended particle structure in stochastic electrodynamics with spin. The correction terms are obtained by considering the internal complex structure of the particle which is a consequence of stochastic average of particle oscillations in the zeropoint field. Hence, the generalised Schrodinger equation may be called stochastic Schrodinger equation. It is found that the second order correction terms are similar to corresponding relativistic corrections. When higher order correction terms are neglected, the stochastic Schrodinger equation reduces to normal Schrodinger equation. It is found that the Schrodinger equation contains an internal structure in disguise and that can be revealed in the form of internal kinetic energy. The internal kinetic energy is found to be equal to the quantum potential obtained in the Madelung fluid theory or Bohm statistical theory. In the rest frame of the particle, the stochastic Schrodinger equation reduces to a Dirac type equation and its Lorentz boost gives the Dirac equation. Finally, the relativistic Klein–Gordon equation is derived by squaring the stochastic Schrodinger equation. The theory elucidates a logical understanding of classical approach to quantum mechanical foundations.

Domain Decomposition Algorithms for Two Dimensional Linear Schrödinger Equation

This paper deals with two domain decomposition methods for two dimensional linear Schrodinger equation, the Schwarz waveform relaxation method and the domain decomposition in space method. After presenting the classical algorithms, we propose a new algorithm for the free Schrodinger equation and a preconditioned algorithm for the general Schrodinger equation. These algorithms are studied numerically, which shows that the two new algorithms could accelerate the convergence and reduce the computation time. Besides the traditional Robin transmission condition, we also propose to use a newly constructed absorbing condition as the transmission condition.

Examination of Schrodinger Equation in Pre-Planckian Space-Time Early Universe

We look at the Vuilli (1999) write up of a generalized Schrodinger equation with its Ricci scalar inclusion, in curved space-time. This has a simplified version in Pre-Planckian regime, which leads to comparing a resultant admissible wave function with Bohmian reformulations of quantum physics. As was done earlier, we compare this result with a formulation of a modified “Poisson” equation from Poissons and Will from 2014, and then use inflaton physics. The resulting inflaton is then compared to the wave functional in the first part of this document.

Kink Formation Dynamics in a Single α-helical Protein

Protein folding remains an open problem despite the progress in understanding the process rate and the success in folding prediction for some small proteins. The reason is the absence of a constructive theoretical framework, both general and specific enough.In earlier papers, we have argued that protein-folding dynamics can be described in terms of solitons of a generalized discrete non-linear Schrodinger equation (GDNLSE) obtained from the energy function in terms of bond and torsion angles [1]. The soliton manifestation is the pattern helix--loop--helix in the secondary structure of the protein, which explains the importance of understanding loop formation in helical proteins and the kink assignment to it [2]. We propose a new mechanism for this process based on the energy transmission along the chain---a disturbance in the latter leads to energy accumulation sufficient to form a kink. We present first insights into the process dynamics by all-atom molecular-dynamics analysis of unfolding of a single alpha-helical protein–--one chain of the core structure of gp41 from the HIV envelope glycoprotein (PDB ID: 1AIK). We suggest an adequate quantification of the side-chain orientation dynamics and also identify some force-field related artefacts.

Sobolev spaces associated with a Dunkl type differential-difference operator on the real line and applications

In this paper we study the Sobolev spaces associated with a Dunkl type differential-difference operator on the real line. As applications we study the theory of reproducing kernels to the Tikhonov regularization, on the generalized Sobolev spaces. We give some properties including some estimates for the solution of the generalized Schrodinger equation.

Process Physics: Emergent Unified Dynamical 3-Space, Quantum and Gravity: a Review

Experiments have repeatedly revealed the existence of a dynamical structured fractal 3-space, with a speed relative to the Earth of some 500 km sec−1 from a southerly direction. Experiments have ranged from optical light speed anisotropy interferometers to zener diode quantum detectors. This dynamical space has been missing from theories from the beginning of physics. This dynamical space generates a growing universe and gravity when included in a generalised Schrodinger equation and light bending when included in generalised Maxwell equations. Here we review ongoing attempts to construct a deeper theory of the dynamical space starting from a stochastic pattern generating model that appears to result in 3-dimensional geometrical elements, “gebits” and accompanying quantum behaviour. The essential concept is that reality is a process and geometrical models for space and time are inadequate.

Generalized Spaces of Sobolev Types and Applications

In this paper, we introduce and we study Sobolev type spaces associated to Jacobi–Cherednik operator on $$\mathbb {R}$$ . Next we define the generalized Besov and Triebel spaces and study some of the properties. As applications on these spaces we establish the Sobolev embedding, the hypoellipticity for the Jacobi–Cherednik operator. We give some properties including some estimates for the solution of the generalized wave equation and the generalized Schrodinger equation. Also, some applications are given for these spaces.

Cherednik-Sobolev spaces and applications

In this paper we introduce and we study the Sobolev type spaces associated to Cherednik operators on \({\mathbb{R }}^{d}\). Next we define the generalized Besov and Triebel spaces and we study some of properties. As applications on these spaces we establish the Sobolev embedding, the hypoellipticity for the Cherednik operators. We give some properties including some estimates for the solution of the generalized wave equation and the generalized Schrodinger equation. Also, some applications are given for these spaces.

Spectral variations in supercontinuum pulse propagation

In this paper, we have used the general nonlinear Schrodinger equation (GNLSE) to study the spectral evolution of a short light pulse propagating through an optical fiber. The equation includes the main factors affecting its propagation through the fiber, involving the linear and nonlinear dispersive characteristics. We find the asymmetry feature of the spectrum due to the Raman Effect. The first order group velocity dispersion sets in after a finite propagation length. We have systematically studied the evolution of the pulse through a nonlinear medium.

Mathematical modeling of quantum well potentials via generalized Darboux transformations

The Darboux transformation operator technique is applied to the generalized Schrodinger equation with a position-dependent effective mass and with linearly energy-dependent potentials. It is shown how to construct the quantum well potentials in nanoelectronic with a given spectrum. The method is illustrated by several examples.

Darboux transformations in integral and differential forms for generalized Schrödinger equations

The Darboux transformation operator technique is applied to the generalized Schrodinger equation with a position-dependent effective mass and with linearly energy-dependent potentials. Intertwining operators are obtained in an explicit form and used for constructing generalized Darboux transformations. An interrelation is established between the differential and integral transformation operators. It is shown how to construct the quantum well potentials in nanoelectronic with a given spectrum.

The effects of minimal length and maximal momentum on the transition rate of ultra cold neutrons in gravitational field

The existence of a minimum observable length and/or a maximum observable momentum is in agreement with various candidates of quantum gravity such as string theory, loop quantum gravity, doubly special relativity and black hole physics. In this scenario, the Heisenberg uncertainty principle is changed to the so-called Generalized (Gravitational) Uncertainty Principle (GUP) which results in modification of all Hamiltonians in quantum mechanics. In this paper, following a recently proposed GUP which is consistent with quantum gravity theories, we study the quantum mechanical systems in the presence of both a minimum length and a maximum momentum. The generalized Hamiltonian contains two additional terms which are proportional to $\alpha p^3$ and $\alpha^2 p^4$ where $\alpha \sim 1/M_{Pl}c$ is the GUP parameter. For the case of a quantum bouncer, we solve the generalized Schrodinger equation in the momentum space and find the modified energy eigenvalues and eigenfunctions up to the second-order in GUP parameter. The effects of the GUP on the transition rate of ultra cold neutrons in gravitational spectrometers are discussed finally.

Family of finite-difference schemes with approximate transparent boundary conditions for the generalized nonstationary Schrödinger equation in a semi-infinite strip

An initial-boundary value problem for the generalized Schrodinger equation in a semi-infinite strip is solved. A new family of two-level finite-difference schemes with averaging over spatial variables on a finite mesh is constructed, which covers a set of finite-difference schemes built using various methods. For the family, an abstract approximate transparent boundary condition (TBC) is formulated and the solutions are proved to be absolutely stable in two norms with respect to both initial data and free terms. A discrete TBC is derived, and the stability of the family of schemes with this TBC is proved. The implementation of schemes with the discrete TBC is discussed, and numerical results are presented.